ON THE CONDUCTION OF HEAT IN CRYSTALS. 209 for which the equation of motion of heat takes the form du A d*u ^d*u ~d*u /KX CO-J- = J. -T-2 + -2-7-2 + ^7-2 ...............(5). r cfa cfe2 dy2 dz* v ' The system of axes of which the existence has just been established may conveniently be called the thermic axes of the crystal. Since the left-hand member of equation (4) is identical with Act? -f- By* + Cz\ it follows that not only do the principal axes of the surface (4) determine the directions of the thermic axes, but the constants A, B, 0, are the squared reciprocals of the principal semi-axes of that surface. 6. Let us now take the thermic axes for axes of coordinates, and investigate the general expression for the flux of heat. The general expressions for fx, fy, fz, being linear functions of the three differential coefficients of u with respect to #, y, z, will con-tain altogether nine arbitrary constants. Substituting the general expressions in (2), and comparing with (5), we find three relations between the constants, depending upon the choice of coordinate axes. These relations being introduced, the expressions may be put under the following form : /• - A^L W ~ J- T? du x~~*-dx ldy+ ^dz , ndu n du w du v-BTy~1}*-dz + ^d~x du du du (6). I shall defer till towards the end of the paper a consideration of the reasons which make it probable that 2)13 EL, Fl are necessarily equal to zero. For the present it may be observed that if the medium be symmetrical with respect to two rectangular planes, these constants must vanish. For the planes of symmetry must evidently contain the thermic axes; and on account of the symmetry supposed, if the planes of symmetry be taken for those of xz and yz, fx must change sign with #, while fy and fz remain unchanged ; and similarly when the sign of y is changed, fy must change sign, while fx and fz remain unchanged. Referring to (6), we see that this requires the constants D15 Ev F1 to vanish, so that du , du , du ,,., S. III.